Finite Group Databasebeta

← #27393819

Subgroups — #27393819

Order 13568.

1114 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811177C211yesmaximal, characteristic
2C2×#16810679C211yesmaximal, characteristic
3#382×D53C211yesmaximal, characteristic
4#20275686C212yesmaximal, complemented
5#20275533C212yesmaximal, complemented
6#20275021C212yesmaximal
7#20275019C212yesmaximal
8#20274731C211yesmaximal, characteristic, complemented
9#20274730C211yesmaximal, characteristic, complemented
10#20274727C211yesmaximal, characteristic, complemented
11C53×#1805C211yesmaximal, characteristic, complemented
12#16811177—24no—
13#16811177C418yescomplemented
14C2×#13133216C2211yescharacteristic
15C2×#13133215C2211yescharacteristic
16C2×#13133207—21no—
17C2×#13133197C2211yescharacteristic
18C2×#13133195—21no—
19C2×#13133193C2211yescharacteristic
20C2×C4×Dic106—21no—
21C2×C4×Dic106C2211yescharacteristic
22C2×#13133174—21no—
23C2×C4⋊C4×D53C2212yes—
24C2×C42×D53C2211yescharacteristic
25#16810679—24no—
26C2×C4⋊#88322C2211yescharacteristic
27C2×#13133142C2212yes—
28C2×#13133141C2212yes—
29C2×#13133140C2212yes—
30C2×#13133129C2212yes—
31C2×#13133127C2212yes—
32#16810489C2211yescharacteristic
33#16810488C2212yes—
34#16810487—21no—
35#16810487—21no—
36#95×D53—21no—
37#95×D53—21no—
38#16810386C2212yes—
39#16810385C2212yes—
40#16810385C2212yes—
41C2×C4×Q8×C53C2211yescharacteristic
42C2×C4⋊C8×C53C2211yescharacteristic
43C2×#103×C53C2212yes—
44C53×#382C2211yescharacteristic
45C53×#340C2212yes—
46C22×Dic106—21no—
47C22×Dic106C2311yescharacteristic
48C22×C4×D53—22no—
49C22×C4×D53D411yescharacteristic
50C22×C4×D53D411yescharacteristic
51C22×C4×D53C2311yescharacteristic
52C22×#88322C2312yes—
53#13133216—22no—
54#13133216C2×C414yescomplemented
55#13133215—22no—
56#13133215C2×C414yescomplemented
57#13133207—28no—
58#13133197—22no—
59#13133197C2×C414yes—
60#13133195—28no—
61#13133193—22no—
62#13133193C2×C414yes—
63C4×Dic106—22no—
64C4×Dic106—28no—
65C4×Dic106C2×C414yescomplemented
66#13133174—28no—
67C2×C4⋊Dic53—21no—
68C2×C4⋊Dic53C2312yes—
69C2×C4⋊Dic53C2311yescharacteristic
70C2×#11719919—21no—
71C2×#11719919—21no—
72C2×#11719919—21no—
73C2×#11719919—21no—
74C2×#11719919—21no—
75C2×#11719919—22no—
76C4⋊C4×D53—42no—
77C4⋊C4×D53—42no—
78C2×#11719921—21no—
79C2×#11719921—21no—
80C2×C4×Dic53—21no—
81C2×C4×Dic53D411yescharacteristic
82C2×C4×Dic53D411yescharacteristic
83C2×C4×Dic53C2311yescharacteristic
84C42×D53C2×C414yes—
85C42×D53D414yes—
86C4⋊#88322—22no—
87#13133142—42no—
88#13133141—42no—
89#13133140—42no—
90#13133129—24no—
91#13133127—24no—
92#13133124—21no—
93#13133124—21no—
94#13133124—22no—
95#13133124—22no—
96C22×Q8×C53C2311yescharacteristic
97C22×C8×C53C2312yes—
98C4×Q8×C53—22no—
99C4×Q8×C53C2×C414yescomplemented
100C2×C4⋊C4×C53—21no—
101C2×C4⋊C4×C53C2312yes—
102C2×C4⋊C4×C53C2311yescharacteristic
103C2×C42×C53—21no—
104C2×C42×C53C2311yescharacteristic
105C4⋊C8×C53—22no—
106#103×C53—42no—
107#95×C53—21no—
108#95×C53—21no—
109C23×D53C2×D411yescharacteristic
110C2×Q8×C53—21no—
111C2×Q8×C53—21no—
112C2×Q8×C53—21no—
113C2×Q8×C53—21no—
114C2×Q8×C53—21no—
115C2×Q8×C53C22×C412yes—
116C22×C4×C53—21no—
117C22×C4×C53—21no—
118C22×C4×C53—22no—
119C22×C4×C53C2×D411yescharacteristic
120C22×C4×C53C2×D411yescharacteristic
121C22×C4×C53C2411yescharacteristic
122C22×Dic53—21no—
123C22×Dic53—21no—
124C22×Dic53—22no—
125C22×Dic53C2×D411yescharacteristic
126C22×Dic53C2×D411yescharacteristic
127C22×Dic53C2×D411yescharacteristic
128C2×C4×D53—24no—
129C2×C4×D53—21no—
130C2×C4×D53—21no—
131C2×C4×D53—21no—
132C2×C4×D53—21no—
133C2×C4×D53—42no—
134C2×C4×D53—44no—
135C2×C4×D53—44no—
136C2×C4×D53—44no—
137C2×C4×D53C22⋊C414yes—
138C2×C4×D53C22⋊C414yes—
139C2×C4×D53C22×C412yes—
140C2×C4×D53C2×D412yes—
141C2×Dic106—21no—
142C2×Dic106—21no—
143C2×Dic106—21no—
144C2×Dic106—21no—
145C2×Dic106—21no—
146C2×Dic106—24no—
147C2×Dic106—44no—
148C2×Dic106C22×C412yes—
149C2×C8×C53—22no—
150C2×C8×C53—22no—
151C2×C8×C53—22no—
152C4⋊C4×C53—24no—
153C4⋊C4×C53—21no—
154C4⋊C4×C53—22no—
155C4⋊C4×C53—22no—
156C4⋊C4×C53C22×C412yes—
157C42×C53—24no—
158C42×C53C22×C412yes—
159C42×C53C2×D412yes—
160#11719921—24no—
161#11719921—24no—
162C4⋊Dic53—21no—
163C4⋊Dic53—24no—
164C4⋊Dic53—22no—
165C4⋊Dic53—22no—
166C4⋊Dic53C22×C412yes—
167#11719919—24no—
168#11719919—24no—
169#11719919—24no—
170#11719919—24no—
171#11719919—24no—
172#11719919—44no—
173C4×Dic53—24no—
174C4×Dic53C22⋊C414yes—
175C4×Dic53C22⋊C414yes—
176C4×Dic53C22×C412yes—
177C4×Dic53C2×D412yes—
178C2×#88322—22no—
179C2×#88322—22no—
180C2×#88322—22no—
181C23×C53C22×D411yescharacteristic
182C22×D53—21no—
183C22×D53—21no—
184C22×D53—22no—
185C22×D53C2×C22⋊C412yes—
186C22×D53#13714yes—
187Q8×C53—22no—
188Q8×C53—22no—
189Q8×C53—41no—
190Q8×C53—41no—
191C2×C4×C53—22no—
192C2×C4×C53—22no—
193C2×C4×C53—22no—
194C2×C4×C53—22no—
195C2×C4×C53—41no—
196C2×C4×C53—41no—
197C2×C4×C53—42no—
198C2×C4×C53—42no—
199C2×C4×C53—42no—
200C2×C4×C53—42no—
201C2×C4×C53C2×C22⋊C412yes—
202C2×C4×C53C2×C22⋊C412yes—
203C2×C4×C53C23×C411yescharacteristic
204C2×C4×C53C22×D411yescharacteristic
205C2×C4×C53H(5,𝔽2)11yescharacteristic
206C2×C4×C53H(5,𝔽2)11yescharacteristic
207C2×C4×C532−1+411yescharacteristic
208C2×C4×C532−1+411yescharacteristic
209C2×Dic53—21no—
210C2×Dic53—21no—
211C2×Dic53—22no—
212C2×Dic53—22no—
213C2×Dic53—22no—
214C2×Dic53—22no—
215C2×Dic53—41no—
216C2×Dic53—41no—
217C2×Dic53—42no—
218C2×Dic53—42no—
219C2×Dic53—42no—
220C2×Dic53—42no—
221C2×Dic53C2×C22⋊C412yes—
222C2×Dic53C2×C22⋊C412yes—
223C2×Dic53C2×C22⋊C412yes—
224C4×D53—28no—
225C4×D53—22no—
226C4×D53—22no—
227C4×D53—22no—
228C4×D53—22no—
229C4×D53—44no—
230C4×D53—88no—
231Dic106—22no—
232Dic106—22no—
233Dic106—48no—
234Dic106—41no—
235Dic106—41no—
236C8×C53—42no—
237#88322—42no—
238#21866—531nomaximal, Sylow
239C22×C53C22×C22⋊C411yescharacteristic
240C22×C53#53511yescharacteristic
241C22×C53#53611yescharacteristic
242C22×C53C2×#13712yes—
243C22×C53#57512yes—
244C2×D53—22no—
245C2×D53—22no—
246C2×D53—24no—
247C2×D53—24no—
248C2×D53#41914yes—
249C4×C53—22no—
250C4×C53—22no—
251C4×C53—42no—
252C4×C53—42no—
253C4×C53—82no—
254C4×C53#52011yescharacteristic
255C4×C53#52011yescharacteristic
256C4×C53#53711yescharacteristic
257C4×C53#53711yescharacteristic
258Dic53—22no—
259Dic53—22no—
260Dic53—22no—
261Dic53—42no—
262Dic53—42no—
263Dic53—82no—
264C2×#533—531no—
265C2×#432—531no—
266C2×#382—531no—
267#1993—532no—
268#1960—532no—
269#1872—532no—
270#1872—532no—
271#1805—531no—
272#1805—531no—
273#1805—531no—
274#1805D5311yescharacteristic, complemented
275C2×C53#231111yescharacteristic
276C2×C53C2×#41911yescharacteristic
277C2×C53#289611yescharacteristic
278C2×C53#321512yes—
279C2×C53#323612yes—
280D53—24no—
281#533—538no—
282#533—1064no—
283C2×C4⋊Q8—531no—
284C2×C4⋊Q8—531no—
285C2×#125—1061no—
286C2×C22⋊Q8—531no—
287C2×C22⋊Q8—531no—
288C2×C22⋊Q8—1061no—
289C2×C4×Q8—531no—
290C2×C4×Q8—1061no—
291C2×C4×Q8C2×D5311yescharacteristic
292C2×#117—1061no—
293C22×C4⋊C4—532no—
294C22×C42—531no—
295#432—1064no—
296C2×C4⋊C8—531no—
297C2×C4⋊C8C2×D5311yescharacteristic
298C2×#103—532no—
299C2×#103—532no—
300C2×#103—532no—
301C2×#103C2×D5312yes—
302C2×C22⋊C8—532no—
303C2×#97—532no—
304#382—532no—
305#382—531no—
306#382—1061no—
307#382—1061no—
308#382C2×D5311yescharacteristic
309C2×#95—1061no—
310C2×#95—1061no—
311#340—532no—
312#340—532no—
313#340—532no—
314#340C2×D5312yes—
315C53#2186611yescharacteristic, complemented, Sylow
316C22×Q8—531no—
317C22×Q8—1061no—
318C22×Q8C22×D5311yescharacteristic
319C23×C4—531no—
320C23×C4—531no—
321C23×C4—531no—
322C23×C4—1062no—
323C22×C8—532no—
324C22×C8C22×D5312yes—
325C4⋊Q8—534no—
326C4⋊Q8—534no—
327C4⋊Q8—1062no—
328C4⋊Q8—1062no—
329#125—1068no—
330C22⋊Q8—534no—
331C22⋊Q8—534no—
332C22⋊Q8—1062no—
333C22⋊Q8—1068no—
334C22⋊Q8—1062no—
335C4×Q8—22no—
336C4×Q8—534no—
337C4×Q8—1068no—
338C4×Q8—1062no—
339C4×Q8C4×D5314yescomplemented
340#117—1068no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—531no—
343C2×C4⋊C4—532no—
344C2×C4⋊C4—1061no—
345C2×C4⋊C4—1061no—
346C2×C4⋊C4—1061no—
347C2×C4⋊C4—1061no—
348C2×C4⋊C4—1061no—
349C2×C4⋊C4—1061no—
350C2×C4⋊C4—1062no—
351C2×C4⋊C4—2122no—
352C2×C4⋊C4—2122no—
353C2×C4⋊C4C22×D5312yes—
354C2×C4⋊C4C22×D5311yescharacteristic
355C2×C22⋊C4—1061no—
356C2×C22⋊C4—1061no—
357C2×C42—21no—
358C2×C42—531no—
359C2×C42—531no—
360C2×C42—531no—
361C2×C42—534no—
362C2×C42—534no—
363C2×C42—1061no—
364C2×C42C22×D5311yescharacteristic
365C4⋊C8—22no—
366C4⋊C8—1062no—
367#103—42no—
368#103—2122no—
369#103—2122no—
370#103—2122no—
371C22⋊C8—1064no—
372#97—1064no—
373#95—21no—
374#95—21no—
375#95—1062no—
376#95—1062no—
377#95—1061no—
378#95—1061no—
379C24—531no—
380C2×Q8—21no—
381C2×Q8—21no—
382C2×Q8—21no—
383C2×Q8—21no—
384C2×Q8—21no—
385C2×Q8—532no—
386C2×Q8—1064no—
387C2×Q8—1061no—
388C2×Q8—1061no—
389C2×Q8—1061no—
390C2×Q8—1061no—
391C2×Q8—1061no—
392C2×Q8—2124no—
393C2×Q8C2×C4×D5312yes—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—531no—
398C22×C4—531no—
399C22×C4—531no—
400C22×C4—534no—
401C22×C4—532no—
402C22×C4—532no—
403C22×C4—534no—
404C22×C4—1064no—
405C22×C4—1061no—
406C22×C4—1061no—
407C22×C4—1061no—
408C22×C4—1061no—
409C22×C4—1061no—
410C22×C4—1061no—
411C22×C4—1062no—
412C22×C4—2122no—
413C22×C4—2124no—
414C22×C4—2124no—
415C22×C4—2124no—
416C22×C4D4×D5311yescharacteristic
417C22×C4D4×D5311yescharacteristic
418C22×C4C23×D5311yescharacteristic
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—1062no—
423C2×C8—1062no—
424C2×C8—1062no—
425C4⋊C4—24no—
426C4⋊C4—21no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—532no—
430C4⋊C4—1064no—
431C4⋊C4—1064no—
432C4⋊C4—1064no—
433C4⋊C4—1064no—
434C4⋊C4—1064no—
435C4⋊C4—1064no—
436C4⋊C4—1061no—
437C4⋊C4—1062no—
438C4⋊C4—1062no—
439C4⋊C4—2124no—
440C4⋊C4C2×C4×D5312yes—
441C22⋊C4—1064no—
442C22⋊C4—1064no—
443C42—24no—
444C42—534no—
445C42—534no—
446C42—532no—
447C42—532no—
448C42—1064no—
449C42C2×C4×D5312yes—
450C42D4×D5312yes—
451C23—532no—
452C23—534no—
453C23—1062no—
454C23—1061no—
455C23—1061no—
456C23C2×D4×D5311yescharacteristic
457Q8—22no—
458Q8—22no—
459Q8—41no—
460Q8—41no—
461Q8—1062no—
462Q8—1062no—
463Q8—2128no—
464Q8—2121no—
465Q8—2121no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—42no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—532no—
477C2×C4—532no—
478C2×C4—532no—
479C2×C4—1068no—
480C2×C4—1062no—
481C2×C4—1062no—
482C2×C4—1061no—
483C2×C4—1061no—
484C2×C4—1062no—
485C2×C4—1062no—
486C2×C4—1062no—
487C2×C4—1062no—
488C2×C4—1062no—
489C2×C4—1062no—
490C2×C4—4248no—
491C2×C4—2124no—
492C2×C4—2122no—
493C2×C4—2122no—
494C2×C4—2122no—
495C2×C4—2122no—
496C2×C4—2121no—
497C2×C4—2121no—
498C2×C4C22⋊C4×D5312yes—
499C2×C4C22⋊C4×D5312yes—
500C2×C4C22×C4×D5311yescharacteristic
501C2×C4C2×D4×D5311yescharacteristic
502C2×C4#1313326211yescharacteristic
503C2×C4#1313326311yescharacteristic
504C2×C4#1313326411yescharacteristic
505C2×C4#1313326511yescharacteristic
506C8—42no—
507C8—2122no—
508C22—534no—
509C22—1064no—
510C22—1062no—
511C22—1062no—
512C22—1064no—
513C22C2×C22⋊C4×D5311yescharacteristic
514C22#1681118911yescharacteristic
515C22#1681119611yescharacteristic
516C22#137×D5312yes—
517C22#1681142012yes—
518C4—22no—
519C4—22no—
520C4—42no—
521C4—42no—
522C4—82no—
523C4—1062no—
524C4—1062no—
525C4—1062no—
526C4—4242no—
527C4—2122no—
528C4—2122no—
529C4#1681110311yescharacteristic
530C4#1681110411yescharacteristic
531C4#1681120211yescharacteristic
532C4#1681120311yescharacteristic
533C2—1064no—
534C2#2027730411yescharacteristic
535C2#419×D5311yescharacteristic
536C2#2028323011yescharacteristic
537C2#2028571612yes—
538C2#2028591912yes—
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1805C2
socleC23×C53C22×D4
prime radicalC23×D53C2×D4
cosocleC22×C4×C53C24
hypercenter#1805D53
hypocenterC53#21866
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1805

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. #20275686C2semidirect
5. #20275533C2semidirect
8. #20274731C2semidirect
9. #20274730C2semidirect
10. #20274727C2semidirect
11. C53×#1805C2semidirect
13. #16811177C4semidirect
54. #13133216C2×C4semidirect
56. #13133215C2×C4semidirect
65. C4×Dic106C2×C4semidirect
99. C4×Q8×C53C2×C4semidirect
274. #1805D53semidirect
315. C53#21866semidirect
339. C4×Q8C4×D53semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#168111772
C2×#168106792
#382×D532
#202756862
#202755332
#202750212
#202750192
#202747312
#202747302
#202747272
C53×#18052
#2186653